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June  2021, 10(2): 249-258. doi: 10.3934/eect.2020064

Decay rate of global solutions to three dimensional generalized MHD system

Yanxia Niu 1, , Yinxia Wang 2,, and  Qingnian Zhang 2,

1. 

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
2. 

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

* Corresponding author: Yinxia Wang

Received  September 2019 Published  June 2021 Early access  June 2020

Fund Project: The second author is supported in part by the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province(Grant No. 20ZX002)

We investigate the initial value problem for the three dimensional generalized incompressible MHD system. Analyticity of global solutions was proved by energy method in the Fourier space and continuous argument. Then decay rate of global small solutions in the function space $ \mathcal {X}^{1-2\alpha}\bigcap \mathcal {X}^{1-2\beta} $ follows by constructing time weighted energy inequality.

Keywords: generalized MHD system, decay rate of global solutions, analyticity.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35B40.
Citation: Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations and Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1.

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.

[6]

J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.

[7]

C. He, X. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.

[12]

Y. Lin, H. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.

[16]

W. Wang, T. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.

[20]

Y. Xiao, B. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1.

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.

show all references

References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1.

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.

[6]

J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.

[7]

C. He, X. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.

[12]

Y. Lin, H. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.

[16]

W. Wang, T. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.

[20]

Y. Xiao, B. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1.

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.

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